Modul matrices

PPR Maths nbk MODULE 12 SKIM TUISYEN FELDA (STF) MATEMATIK SPM “ENRICHMENT” TOPIC : MATRICES TIME : 2 HOURS

1.

(a)

⎛3 − 2⎞ ⎟⎟ is m ⎝5 − 4⎠

The inverse matrix of ⎜⎜

⎛− 4 ⎜⎜ ⎝− 5

n⎞ ⎟ 3 ⎟⎠

Find the value of m and of n. (b)

Answer : (a)

(b)

Hence, using matrices, solve the following simultaneous equations : 3x – 2y = 8 5x – 4y = 13

PPR Maths nbk 2.

(a)

⎛ m 3⎞ ⎟⎟ 2 n ⎝ ⎠

Given that G = ⎜ ⎜

and the inverse matrix of G is

1 ⎛ 4 − 3⎞ ⎜ ⎟, 14 ⎜⎝ − 2 m ⎟⎠

find the value of m and of n. (b)

Hence, using matrices, calculate the value of p and of q that satisfies the following equation :

⎛ p⎞ ⎛ 1 ⎞ G⎜⎜ ⎟⎟ = ⎜⎜ ⎟⎟ ⎝ q ⎠ ⎝ − 8⎠ Answer : (a)

(b)

PPR Maths nbk 3.

⎛ −1 2⎞ ⎛1 0⎞ ⎟⎟ = ⎜⎜ ⎟⎟, A⎜⎜ − 3 5 0 1 ⎝ ⎠ ⎝ ⎠

(a)

Given that

(b)

Hence, using the matrix method, find the value of r and s which satisfy the simultaneous equations below.

-r + 2s = -4 -3r + 5s = -9

Answer : (a)

(b)

find matrix A.

PPR Maths nbk 4.

⎛ 4 5⎞ ⎟⎟ and matrix PQ = 6 8 ⎝ ⎠

Given matrix P = ⎜ ⎜ (a) (b)

Answer : (a)

(b)

⎛1 0⎞ ⎜⎜ ⎟⎟ 0 1 ⎝ ⎠

Find the matrix Q. Hence, calculate by using the matrix method, the values of m and n that satisfy the following simultaneous linear equations : 4m + 5n = 7 6m + 8n = 10

PPR Maths nbk 5.

⎛ 4 − 3⎞ ⎟⎟ , 8 − 5 ⎝ ⎠

Given the matrix P is ⎜ ⎜

⎛1 0⎞ ⎟⎟ 0 1 ⎝ ⎠

(a)

Find the matrix Q so that PQ = ⎜⎜

(b)

Hence, calculate the values of h and k, which satisfy the matrix equation:

⎛ 4 − 3 ⎞⎛ h ⎞ ⎛ − 7 ⎞ ⎜⎜ ⎟⎟⎜⎜ ⎟⎟ = ⎜⎜ ⎟⎟ 8 − 5 k − 11 ⎝ ⎠⎝ ⎠ ⎝ ⎠ Answer : (a)

(b)

PPR Maths nbk 6.

⎛ k 6⎞ ⎟⎟, find the value of k if matrix M has no inverse. − 4 2 ⎝ ⎠

(a)

Given matrix M = ⎜ ⎜

(b)

Given the matrix equations

⎛ 7 − 6 ⎞⎛ x ⎞ ⎛ − 4 ⎞ ⎜⎜ ⎟⎟⎜⎜ ⎟⎟ = ⎜⎜ ⎟⎟ ⎝ − 5 8 ⎠⎝ y ⎠ ⎝ 1 ⎠ (i) (ii) Answer : (a)

(b)

⎛ x⎞

1 ⎛ 8 6 ⎞⎛ − 4 ⎞

and ⎜ ⎟ = ⎜ ⎜ y ⎟ h ⎜ 5 7 ⎟⎟⎜⎜ 1 ⎟⎟ ⎝ ⎠ ⎝ ⎠⎝ ⎠

Find the value of h Hence, find the value of x and y.

PPR Maths nbk 7.

⎛2 5 ⎞ ⎟⎟ does not have an inverse matrix. k − 2 ⎝ ⎠

It is given that matrix P = ⎜ ⎜ (a) (b)

Find the value of k.

If k = 1, find the inverse matrix of P and hence, using matrices, find the values of x and y that satisfy the following simultaneous linear equations.

2x + 5y = 13 x - 2y = -7

Answer : (a)

(b)

PPR Maths nbk 8.

⎛ 2 4⎞ ⎛ 2 4⎞ ⎟⎟ M = ⎜⎜ ⎟⎟ 1 3 1 3 ⎝ ⎠ ⎝ ⎠

(a)

Find matrix M such that ⎜ ⎜

(b)

Using matrices, calculate the values of x and y that satisfy the following matrix equation.

⎛ 2 4 ⎞⎛ x ⎞ ⎛ 6 ⎞ ⎜⎜ ⎟⎟⎜⎜ ⎟⎟ = ⎜⎜ ⎟⎟ ⎝ 1 3 ⎠⎝ y ⎠ ⎝ 5 ⎠

Answer : (a)

(b)

PPR Maths nbk

9.

⎛3 −1⎞ ⎟⎟ .

5 − 2 ⎝ ⎠

(a)

Find the inverse of matrix ⎜ ⎜

(b)

Hence, using matrices, calculate the values of d and e that satisfy the following simultaneous equations : 2d – e = 7 5d – e = 16

Answer : (a)

(b)

PPR Maths nbk

10.

⎛1 − 2⎞ ⎟⎟ , find 2 5 ⎝ ⎠

Given matrix M = ⎜ ⎜ (a) (b)

the inverse matrix of M hence, using matrices, the values of u and v that satisfy the following simultaneous equations : u – 2v = 8 2u + 5v = 7

Answer : (a)

(b)

PPR Maths nbk MODULE 12 - ANSWERS TOPIC : MATRICES

1.

m= −

(a)

1 2

1m

n =2 (b) ⎛ 3

⎜⎜ ⎝5

1m

− 2⎞ ⎛ x ⎞ ⎛8 ⎞ ⎟⎜ ⎟ = ⎜ ⎟ − 4 ⎟⎠ ⎜⎝ y ⎟⎠ ⎜⎝13 ⎟⎠ ⎛ x ⎞ 1 ⎛− 4 ⎜⎜ ⎟⎟ = ⎜⎜ ⎝ y⎠ 2 ⎝− 5 x=3 y= −

2.

(a)

⎜⎜ ⎝2

2 ⎞⎛ 8 ⎞ ⎟⎜ ⎟ 3 ⎟⎠⎜⎝13 ⎟⎠

1 2

1m

1m 1m

3 ⎞⎛ p ⎞ ⎟⎜ ⎟ = 4 ⎟⎠⎜⎝ q ⎟⎠

⎛ 1 ⎞ ⎜⎜ ⎟⎟ ⎝ − 8⎠

⎛ p ⎞ 1 ⎛ 4 − 3 ⎞⎛ 1 ⎞ ⎜⎜ ⎟⎟ = ⎜⎜ ⎟⎟⎜⎜ ⎟⎟ ⎝ q ⎠ 14 ⎝ − 2 5 ⎠⎝ − 8 ⎠ p=2 q = -3

3.

(a)

(b)

4.

(a)

⎛5 ⎜3 ⎝

A=⎜

1m

1m

n =4 m=5

(b) ⎛ 5

1m

1m

1m

1m 1m

− 2⎞ ⎟ − 1 ⎟⎠

2m

⎛ −1 2⎞ ⎛ r ⎞ ⎛ − 4⎞ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ = ⎜⎜ ⎟⎟ ⎝− 3 5⎠ ⎝ s ⎠ ⎝ − 9⎠

1m

⎛ r ⎞ 1 ⎛ 5 − 2 ⎞⎛ − 4 ⎞ ⎜⎜ ⎟⎟ = ⎜⎜ ⎟⎟⎜⎜ ⎟⎟ ⎝ s ⎠ 1 ⎝ 3 − 1 ⎠⎝ − 9 ⎠

1m

P=

r = -2

1m

s = -3

1m 1m

1 ⎛ 8 − 5⎞ ⎜ ⎟ 32 − 30 ⎜⎝ − 6 4 ⎟⎠

PPR Maths nbk

(b)

5.

(a)

(b)

6.

1 ⎛ 8 − 5⎞ ⎜ ⎟ = 2 ⎜⎝ − 6 4 ⎟⎠

1m

⎛ 4 5 ⎞⎛ m ⎞ ⎛ 7 ⎞ ⎜⎜ ⎟⎟⎜⎜ ⎟⎟ = ⎜⎜ ⎟⎟ ⎝ 6 8 ⎠⎝ n ⎠ ⎝10 ⎠

1m

⎛ m ⎞ 1 ⎛ 8 − 5 ⎞⎛ 7 ⎞ ⎜⎜ ⎟⎟ = ⎜⎜ ⎟⎟⎜⎜ ⎟⎟ ⎝ n ⎠ 2 ⎝ − 6 4 ⎠⎝10 ⎠

1m

m=3

1m

n = -1

1m

⎛ − 5 3 ⎞ 1m 1 ⎜ ⎟ − 20 − (−24) ⎜⎝ 8 4 ⎟⎠ 1 ⎛ − 5 3⎞ 1m ⎟ = ⎜⎜ 4 ⎝ 8 4 ⎟⎠

P =

⎛ 4 − 3 ⎞⎛ h ⎞ ⎛ − 7 ⎞ ⎜⎜ ⎟⎟⎜⎜ ⎟⎟ = ⎜⎜ ⎟⎟ ⎝ 8 − 5 ⎠⎝ k ⎠ ⎝ − 11⎠ ⎛ h ⎞ 1 ⎛ − 5 3 ⎞⎛ − 7 ⎞ ⎜⎜ ⎟⎟ = ⎜⎜ ⎟⎟⎜⎜ ⎟⎟ ⎝ k ⎠ 2 ⎝ 8 4 ⎠⎝ − 11⎠ 1⎛ 2 ⎞ ⎟ = ⎜⎜ 2 ⎝ − 100 ⎟⎠

1m

1m

h=1 k = -50

1m 1m

(a)

k = -12

1m

(b)

(i)

h = 26

⎛ x⎞ 1 ⎛ 8 6 ⎞⎛ − 4 ⎞ ⎜⎜ ⎟⎟ = ⎜⎜ ⎟⎟⎜⎜ ⎟⎟ ⎝ y ⎠ 26 ⎝ 5 7 ⎠⎝ 1 ⎠ 1 ⎛ − 26 ⎞ ⎜ ⎟ = 26 ⎜⎝ − 13 ⎟⎠

1m

PPR Maths nbk (ii)

1m

1m

x = -1 y= −

7.

(a)

1m

1 2

- 4 – 5k = 0

1m

1m

5k = -4 k= − (b)

8.

4 5

1m

⎛ 2 5 ⎞⎛ x ⎞ ⎛ 13 ⎞ ⎜⎜ ⎟⎟⎜⎜ ⎟⎟ = ⎜⎜ ⎟⎟ ⎝ 1 − 2 ⎠⎝ y ⎠ ⎝ − 7 ⎠

1m

⎛ x⎞ 1 ⎛ − 2 − 5 ⎞⎛ 13 ⎞ ⎜⎜ ⎟⎟ = − ⎜⎜ ⎟⎟⎜⎜ ⎟⎟ 1 2 − y 9 ⎝ ⎠ ⎝ ⎠⎝ − 7 ⎠

1m

x = -1 y=3

1m 1m

⎛1 0⎞ ⎟⎟ ⎝0 1⎠

(a)

M= ⎜ ⎜

2m

(b)

⎛ x⎞ 1 ⎛ 3 − 4 ⎞⎛ 6 ⎞ ⎜⎜ ⎟⎟ = ⎜⎜ ⎟⎟⎜⎜ ⎟⎟ ⎝ y ⎠ 6 − 4 ⎝ − 1 2 ⎠⎝ 5 ⎠

1m

1 ⎛ 3 − 4 ⎞⎛ 6 ⎞ ⎜ ⎟⎜ ⎟ 2 ⎜⎝ − 1 2 ⎟⎠⎜⎝ 5 ⎟⎠ 1 ⎛ − 2⎞ = ⎜⎜ ⎟⎟ 2⎝ 4 ⎠

=

x = -1 y=2

1m

1m 1m

PPR Maths nbk 9.

(a)

1 ⎛ − 2 1⎞ ⎜ ⎟ − 6 + 5 ⎜⎝ − 5 3 ⎟⎠

1m

1 ⎛ − 2 1⎞ ⎜ ⎟ − 1 ⎜⎝ − 5 3 ⎟⎠

1m

=

− 1 ⎞⎛ d ⎞ ⎛ 7 ⎞ ⎜⎜ ⎟⎟⎜⎜ ⎟⎟ = ⎜⎜ ⎟⎟ ⎝ 5 − 3 ⎠⎝ e ⎠ ⎝16 ⎠ ⎛ d ⎞ 1 ⎛ − 3 1 ⎞⎛ 7 ⎞ ⎜⎜ ⎟⎟ = ⎜⎜ ⎟⎟⎜⎜ ⎟⎟ ⎝ e ⎠ − 1 ⎝ − 5 2 ⎠⎝16 ⎠

(b) ⎛ 2

=

1m

1m

1 ⎛ − 5⎞ ⎜ ⎟ − 1 ⎜⎝ − 3 ⎟⎠

⎛ 5⎞ = ⎜⎜ ⎟⎟ ⎝ 3⎠ d=5 e=3

10.

(a)

⎛ 5 2⎞ 1 ⎜ ⎟ 5 − (−4) ⎜⎝ − 2 1 ⎟⎠ 1 ⎛ 5 2⎞ ⎟ = ⎜⎜ 9 ⎝ − 2 1 ⎟⎠

− 2 ⎞⎛ u ⎞ ⎛ 8 ⎞ ⎜ 2 5 ⎟⎟⎜⎜ v ⎟⎟ = ⎜⎜ 7 ⎟⎟ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎛ u ⎞ 1 ⎛ 5 2 ⎞⎛ 8 ⎞ ⎟⎟⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ = ⎜⎜ ⎝ v ⎠ 9 ⎝ − 2 1 ⎠⎝ 7 ⎠

(b) ⎛⎜ 1

1m 1m

1m

1m

1m

1m

1 ⎛ 54 ⎞ = ⎜⎜ ⎟⎟ 9 ⎝ − 9⎠ ⎛6⎞ = ⎜⎜ ⎟⎟ ⎝ − 1⎠ u=6 v = −1

1m 1m